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January 2000 Long-range dependence and Appell rank
Donatas Surgailis
Ann. Probab. 28(1): 478-497 (January 2000). DOI: 10.1214/aop/1019160127


We study limit distributions of sums $S_N^{(G)} = \sum_{t=1}^N G(X_t)$ of nonlinear functions $G(x)$ in stationary variables of the form $X_t = Y_t + Z_t$, where ${Y_t}$ is a linear (moving average) sequence with long-range dependence, and ${Z_ t}$ is a (nonlinear) weakly dependent sequence. In particular, we consider the case when ${Y_ t}$ is Gaussian and either (1)${Z_t}$ is a weakly dependent multilinear form in Gaussian innovations, or (2) ${Z_t}$ is a finitely dependent functional in Gaussian innovations or (3)${Z_t}$ is weakly dependent and independent of $Y_t$ . We show in all three cases that the limit distribution of $S^(G)_N$ is determined by the Appell rank of $G( x)$, or the lowest $k\geq 0$ such that $a_k = \partial^k E\{G(X_0+c)\}/\partial c^k|_{c=0 \not= 0$.


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Donatas Surgailis. "Long-range dependence and Appell rank." Ann. Probab. 28 (1) 478 - 497, January 2000.


Published: January 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1130.60306
MathSciNet: MR1756013
Digital Object Identifier: 10.1214/aop/1019160127

Primary: 60F05
Secondary: 60G15 , 62M10

Keywords: Appell polynomials , Appell rank , long-range dependence , noncentral limit theorems , reduction principle

Rights: Copyright © 2000 Institute of Mathematical Statistics

Vol.28 • No. 1 • January 2000
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